Inversion Sum

AtCoder

IDagc030_d

Time3000ms

Memory256MB

Difficulty—

You are given an integer sequence of length $N$: $A_1,A_2,...,A_N$. Let us perform $Q$ operations in order. The $i$\-th operation is described by two integers $X_i$ and $Y_i$. In this operation, we will choose one of the following two actions and perform it: * Swap the values of $A_{X_i}$ and $A_{Y_i}$ * Do nothing There are $2^Q$ ways to perform these operations. Find the sum of the inversion numbers of the final sequences for all of these ways to perform operations, modulo $10^9+7$. Here, the inversion number of a sequence $P_1,P_2,...,P_M$ is the number of pairs of integers $(i,j)$ such that $1\leq i < j\leq M$ and $P_i > P_j$. ## Constraints * $1 \leq N \leq 3000$ * $0 \leq Q \leq 3000$ * $0 \leq A_i \leq 10^9(1\leq i\leq N)$ * $1 \leq X_i,Y_i \leq N(1\leq i\leq Q)$ * $X_i\neq Y_i(1\leq i\leq Q)$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $Q$ $A_1$ $:$ $A_N$ $X_1$ $Y_1$ $:$ $X_Q$ $Y_Q$ [samples]

Samples

Input #1

Output #1

6

There are four ways to perform operations, as follows:

*   Do nothing, both in the first and second operations. The final sequence would be $1,2,3$, with the inversion number of $0$.
*   Do nothing in the first operation, then perform the swap in the second. The final sequence would be $3,2,1$, with the inversion number of $3$.
*   Perform the swap in the first operation, then do nothing in the second. The final sequence would be $2,1,3$, with the inversion number of $1$.
*   Perform the swap, both in the first and second operations. The final sequence would be $3,1,2$, with the inversion number of $2$.

The sum of these inversion numbers, $0+3+1+2=6$, should be printed.

Input #2

Output #2

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Inversion Sum",
    "description": {
      "content": "You are given an integer sequence of length $N$: $A_1,A_2,...,A_N$. Let us perform $Q$ operations in order. The $i$\\-th operation is described by two integers $X_i$ and $Y_i$. In this operation, we wi",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc030_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given an integer sequence of length $N$: $A_1,A_2,...,A_N$. Let us perform $Q$ operations in order. The $i$\\-th operation is described by two integers $X_i$ and $Y_i$. In this operation, we wi...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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